.\" Copyright 2002 Walter Harms (walter.harms@informatik.uni-oldenburg.de)
.\"
.\" SPDX-License-Identifier: GPL-1.0-or-later
.\"
.TH complex 7 2024-06-15 "Linux man-pages 6.9.1"
.SH NAME
complex \- basics of complex mathematics
.SH LIBRARY
Math library
.RI ( libm ", " \-lm )
.SH SYNOPSIS
.nf
.B #include <complex.h>
.fi
.SH DESCRIPTION
Complex numbers are numbers of the form z = a+b*i, where a and b are
real numbers and i = sqrt(\-1), so that i*i = \-1.
.P
There are other ways to represent that number.
The pair (a,b) of real
numbers may be viewed as a point in the plane, given by X- and
Y-coordinates.
This same point may also be described by giving
the pair of real numbers (r,phi), where r is the distance to the origin O,
and phi the angle between the X-axis and the line Oz.
Now
z = r*exp(i*phi) = r*(cos(phi)+i*sin(phi)).
.P
The basic operations are defined on z = a+b*i and w = c+d*i as:
.TP
.B addition: z+w = (a+c) + (b+d)*i
.TP
.B multiplication: z*w = (a*c \- b*d) + (a*d + b*c)*i
.TP
.B division: z/w = ((a*c + b*d)/(c*c + d*d)) + ((b*c \- a*d)/(c*c + d*d))*i
.P
Nearly all math function have a complex counterpart but there are
some complex-only functions.
.SH EXAMPLES
Your C-compiler can work with complex numbers if it supports the C99 standard.
The imaginary unit is represented by I.
.P
.EX
/* check that exp(i * pi) == \-1 */
#include <math.h>        /* for atan */
#include <stdio.h>
#include <complex.h>
\&
int
main(void)
{
    double pi = 4 * atan(1.0);
    double complex z = cexp(I * pi);
    printf("%f + %f * i\[rs]n", creal(z), cimag(z));
}
.EE
.SH SEE ALSO
.BR cabs (3),
.BR cacos (3),
.BR cacosh (3),
.BR carg (3),
.BR casin (3),
.BR casinh (3),
.BR catan (3),
.BR catanh (3),
.BR ccos (3),
.BR ccosh (3),
.BR cerf (3),
.BR cexp (3),
.BR cexp2 (3),
.BR cimag (3),
.BR clog (3),
.BR clog10 (3),
.BR clog2 (3),
.BR conj (3),
.BR cpow (3),
.BR cproj (3),
.BR creal (3),
.BR csin (3),
.BR csinh (3),
.BR csqrt (3),
.BR ctan (3),
.BR ctanh (3)
